We study the Ising phase transition on a three-dimensional cubic lattice augmented with sparse, distance-dependent shortcuts whose density decays as P (r) proportional to r^ (-2). This network ensemble, characterized by geometric embedding, sparse additive quenched shortcuts, and power-law distance dependence, falls outside the scope of existing results for both uniform small-world networks (Barrat-Weigt, Herrero) and dense long-range Ising models (Fisher-Ma-Nickel, Sak). We present partial analytical results. The effective propagator exhibits three regimes: a Wilson-Fisher intermediate regime (standard Laplacian dominant), a Levy IR regime (wormhole term dominant), and a biharmonic UV regime, with propagator crossover at approximately 125 lattice units. A scaling estimate suggests the nonlocal operator is only marginally relevant at Wilson-Fisher, implying a potentially very slow RG flow. A mean free path analysis yields a finite effective signal velocity of approximately 2. 4 times the local propagation speed, with the exponent alpha = 2 lying at the apparent boundary for finite-speed propagation on sparse shortcut networks. We propose a concrete Monte Carlo simulation protocol (approximately 500 CPU-hours) using Wolff cluster algorithm and finite-size scaling with Binder cumulant analysis to definitively determine the universality class. The key observable is U4*, which unambiguously distinguishes 3D Ising (U4* approximately 0. 61) from mean-field (U4* approximately 0. 47) universality. Control simulations on pure 3D lattice and uniform shortcut networks are specified. The problem is relevant to discrete quantum gravity models with holographic connectivity, network science, and the theory of phase transitions on complex geometries. All analytical results are clearly labeled as scaling estimates or conjectures where appropriate, and five explicit unresolved issues are identified.
Štěpán Sekanina (2026) studied this question.